1cm introduction to channel coding -0 - iit hyderabad
TRANSCRIPT
Introduction to channelcodingShashank Vatedka
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What is the maximum rate at which we canreliably communicate across a noisy channel?
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Not just cellular...
§ WiFi/deep space/wireless
§ Wireline/optical
§ Storage
Very general!
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Digital Communication system
Source signal
Sampling
A/D convertSourcecoding
Channelcoding
ModulatorD/A
Demodulator
A/D
Channeldecoder
Sourcedecoder
Reconstruction
D/A convert
Decodedsignal
Noisycommunication
medium
Effective discrete "channel"
Transmitter
Receiver
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What is the maximum rate at which we canreliably communicate across a discrete
memoryless channel?
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Discrete memoryless channel
ChannelEncoder Decoder
§ Mk „ iid Unif(t0,1uk)§ Memoryless channel:
pY n|Xnpyn|xnq “n
ź
i“1pY |X pyi |xiq
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Discrete memoryless channel
ChannelEncoder Decoder
§ Mk „ iid Unif(t0,1uk)§ Memoryless channel:
pY n|Xnpyn|xnq “n
ź
i“1pY |X pyi |xiq
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Common channels
Binary symmetric channel: BSC(p)
X “ Y “ t0,1u, and
pY |X py|xq “#
1´ p if x “ yp if x ‰ y.
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Common channels
Binary erasure channel: BEC(p)
X “ t0,1u, Y “ t0,1,eu
pY |X py|xq “
$
’
&
’
%
p if y “ e1´ p if x “ y0 otherwise.
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Common channels
Additive white Gaussian noise (AWGN) channel
X “ Y “ RYi “ xi ` Zi , i “ 1,2, . . . ,n
where pZ1, . . . ,Znq are iid with N p0, σ2q components.Power constraint:
}xn}2def“
nÿ
i“1x2
i ď nP
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Common channels
Complex slow/quasi-static fading channel
X “ Y “ CYi “ hXi ` Zi ,
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Common channels
Fast fading channel
Yi “ hiXi ` Zi ,
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Common channels
Multiple antenna/multi-input multi-output (MIMO) channels
X “ Rts , Y “ Rtr .
Y i “ HiX i ` Z i , i “ 1, . . . ,n
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Common channels
A simple channel with memory
Yi “ a0Xi ` a1Xi´1 ` . . .` akXi´k ` Zi
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Common channelsInsertion/deletion channels
2 2 1 1 0 1 0 2 2 1
2 2 1 1 1 0 1 0 2 2 1
insertion
Tx
Rx
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Channel codes
ChannelEncoder Decoder
§ Encoder: f : t0,1uk Ñ X n
§ Decoder: g : Yn Ñ t0,1uk§ Rate:
R “ kn
§ Probability of error:
Pe “ Prr pMk ‰ Mks
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Mutual information
IpX ; Y q def“
ÿ
xPX ,yPYpXY px, yq log2
pXY px, yqpX pxqpY pyq
,
§ Mutual information is symmetric
§ Measures the information that X gives about Y , or Y givesabout X .
§ What happens if X and Y are independent?
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Mutual information
IpX ; Y q def“
ÿ
xPX ,yPYpXY px, yq log2
pXY px, yqpX pxqpY pyq
,
§ Mutual information is symmetric
§ Measures the information that X gives about Y , or Y givesabout X .
§ What happens if X and Y are independent?
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Channel capacity
Maximum rate R for which limnÑ8 Pe “ 0.
Theorem (Shannon)
C “ maxpX
IpX ; Y q.
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A brief history of channelcoding
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The early codes
§ BCH, Reed Solomon, Reed Muller codes (1960s)
§ Convolutional codes (1955-1967)
§ Concatenated codes (1966): deep space communication
§ Trellis coded modulation (1982?): telephone lines
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Performance
0Costello and Forney, “Channel Coding: The Road to Channel Capacity,”Proceedings of the IEEE, 2007. Link:https://arxiv.org/pdf/cs/0611112.pdf 15 / 18
Performance
0Costello and Forney, “Channel Coding: The Road to Channel Capacity,”Proceedings of the IEEE, 2007. Link:https://arxiv.org/pdf/cs/0611112.pdf 16 / 18
Modern codes
§ Turbo codes (1993)
§ LDPC codes (Gallager 1960, rediscovered 2000s)
§ Polar codes (2009)
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Performance
0Costello and Forney, “Channel Coding: The Road to Channel Capacity,”Proceedings of the IEEE, 2007. Link:https://arxiv.org/pdf/cs/0611112.pdf 18 / 18
Performance
0Costello and Forney, “Channel Coding: The Road to Channel Capacity,”Proceedings of the IEEE, 2007. Link:https://arxiv.org/pdf/cs/0611112.pdf 19 / 18